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      Questions tagged [cv.complex-variables]

      Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

      1
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      0answers
      90 views

      Affine algebraic variety as a set of common zeroes of holomorphic functions on ${\mathbb C}^n$

      Let $V$ be an affine algebraic variety in ${\mathbb C}^n$, i.e. a set of common zeroes of some set $S$ of polynomials on ${\mathbb C}^n$: $$ V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}. $$ ...
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      votes
      2answers
      195 views

      Sums of entire surjective functions

      Suppose $(f_n)_n$ is a countable family of entire, surjective functions, each $f_n:\mathbb{C}\to\mathbb{C}$. Can one always find complex scalars $(a_n)_n$, not all zero, such that $\sum_{n=1}^{\...
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      votes
      3answers
      138 views

      Existence of solution to linear fractional equation

      We consider the equation $$?\sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$ where $\lambda_j>0$ and $x_j$ are real distinct numbers. I want to show that if $\lambda_k$ is small compared to the ...
      6
      votes
      1answer
      162 views

      Do analytic functionals form a cosheaf?

      Let $X$ be a complex-analytic manifold. Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
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      votes
      0answers
      45 views

      What is the Laurent series of z+(1/z)? [closed]

      What is the Laurent series of z+(1/z)? Is it just the series itself?
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      votes
      0answers
      54 views

      Metric with singularities on Riemann Surfaces and the associated Laplacians

      I have asked this question on Math Stack Exchange Metric with singularities and associated Laplacian but I have not got any answers/comments, therefore I post this question on the MO. Suppose $M$ ...
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      votes
      0answers
      32 views

      Is there a way to categorise the valleys of a holomorphic function (potentially of $\geq 2$ variables) (multidimensional steepest descent)

      More specifically, I am particularly interested in the question: given some $f:\mathbb{C}^n \to \mathbb{C}$, can we categorise $\mathbb{C}^n$ by which valley the steepest descent curves of a point (...
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      votes
      1answer
      197 views

      Dependence of a solution of a linear ODE on parameter

      Is the following theorem known, or can be easily derived from known results? Consider the differential equation $$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$ where $k>0$ is fixed, $\lambda$ is a large (...
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      votes
      0answers
      106 views

      Is there any neat way to calculate the Fourier transform for an inverse Vandermonde determinant?

      $$\mathrm{PV}\int_{\mathbb R^n} \frac{e^{-i\langle w, x\rangle}}{\prod_{j<k}(x_k-x_j)}dx=?$$ Other than integrate this term by term (which might look crazy)? Let $f(x)=1/\prod_{j<k}(x_k-x_j)$, ...
      3
      votes
      2answers
      378 views

      Reference request: Oldest complex analysis books with (unsolved) exercises?

      Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ...
      1
      vote
      1answer
      116 views

      An equation with Gamma Euler function in critical strip

      Let $$ D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \} $$ that is the critical strip without critical line. I have to find if the following equation, with ...
      3
      votes
      0answers
      103 views

      Injective resolution of the ring of entire functions

      Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis. I would think that the injective dimension ...
      3
      votes
      1answer
      48 views

      Rational approximation on rotation invariant compact subsets of complex plane

      What does the Vitushkin's theorem say about the equality $A(K) = R(K)$ in the special case when $K$ is rotation invariant? More precisely, what are necessary and/or sufficient conditions on $\{|k|: k \...
      2
      votes
      2answers
      106 views

      Coefficients of entire functions with specified zero set

      Let $Z \subseteq \mathbb{C}$ without limit point. By the Weierstrass factorization theorem there is an entire function $h$ those zero set is $Z$. Let $a_n > 0$ be a sequence where $\lim_n \sqrt[n]{...
      3
      votes
      1answer
      160 views

      Local phase statistics of the nontrivial Riemann zeros

      (The question is inspired by Owen Maresh's post) The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$. Numerical results on the first 10000 zeros suggest ...

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